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Simplifying (2x2 + 8x + -3)(2x + -7) + -1(x2 + -7x + 16)(4x + 8) = 0 Reorder the terms: (-3 + 8x + 2x2)(2x + -7) + -1(x2 + -7x + 16)(4x + 8) = 0 Reorder the terms: (-3 + 8x + 2x2)(-7 + 2x) + -1(x2 + -7x + 16)(4x + 8) = 0 Multiply (-3 + 8x + 2x2) * (-7 + 2x) (-3(-7 + 2x) + 8x * (-7 + 2x) + 2x2 * (-7 + 2x)) + -1(x2 + -7x + 16)(4x + 8) = 0 ((-7 * -3 + 2x * -3) + 8x * (-7 + 2x) + 2x2 * (-7 + 2x)) + -1(x2 + -7x + 16)(4x + 8) = 0 ((21 + -6x) + 8x * (-7 + 2x) + 2x2 * (-7 + 2x)) + -1(x2 + -7x + 16)(4x + 8) = 0 (21 + -6x + (-7 * 8x + 2x * 8x) + 2x2 * (-7 + 2x)) + -1(x2 + -7x + 16)(4x + 8) = 0 (21 + -6x + (-56x + 16x2) + 2x2 * (-7 + 2x)) + -1(x2 + -7x + 16)(4x + 8) = 0 (21 + -6x + -56x + 16x2 + (-7 * 2x2 + 2x * 2x2)) + -1(x2 + -7x + 16)(4x + 8) = 0 (21 + -6x + -56x + 16x2 + (-14x2 + 4x3)) + -1(x2 + -7x + 16)(4x + 8) = 0 Combine like terms: -6x + -56x = -62x (21 + -62x + 16x2 + -14x2 + 4x3) + -1(x2 + -7x + 16)(4x + 8) = 0 Combine like terms: 16x2 + -14x2 = 2x2 (21 + -62x + 2x2 + 4x3) + -1(x2 + -7x + 16)(4x + 8) = 0 Reorder the terms: 21 + -62x + 2x2 + 4x3 + -1(16 + -7x + x2)(4x + 8) = 0 Reorder the terms: 21 + -62x + 2x2 + 4x3 + -1(16 + -7x + x2)(8 + 4x) = 0 Multiply (16 + -7x + x2) * (8 + 4x) 21 + -62x + 2x2 + 4x3 + -1(16(8 + 4x) + -7x * (8 + 4x) + x2(8 + 4x)) = 0 21 + -62x + 2x2 + 4x3 + -1((8 * 16 + 4x * 16) + -7x * (8 + 4x) + x2(8 + 4x)) = 0 21 + -62x + 2x2 + 4x3 + -1((128 + 64x) + -7x * (8 + 4x) + x2(8 + 4x)) = 0 21 + -62x + 2x2 + 4x3 + -1(128 + 64x + (8 * -7x + 4x * -7x) + x2(8 + 4x)) = 0 21 + -62x + 2x2 + 4x3 + -1(128 + 64x + (-56x + -28x2) + x2(8 + 4x)) = 0 21 + -62x + 2x2 + 4x3 + -1(128 + 64x + -56x + -28x2 + (8 * x2 + 4x * x2)) = 0 21 + -62x + 2x2 + 4x3 + -1(128 + 64x + -56x + -28x2 + (8x2 + 4x3)) = 0 Combine like terms: 64x + -56x = 8x 21 + -62x + 2x2 + 4x3 + -1(128 + 8x + -28x2 + 8x2 + 4x3) = 0 Combine like terms: -28x2 + 8x2 = -20x2 21 + -62x + 2x2 + 4x3 + -1(128 + 8x + -20x2 + 4x3) = 0 21 + -62x + 2x2 + 4x3 + (128 * -1 + 8x * -1 + -20x2 * -1 + 4x3 * -1) = 0 21 + -62x + 2x2 + 4x3 + (-128 + -8x + 20x2 + -4x3) = 0 Reorder the terms: 21 + -128 + -62x + -8x + 2x2 + 20x2 + 4x3 + -4x3 = 0 Combine like terms: 21 + -128 = -107 -107 + -62x + -8x + 2x2 + 20x2 + 4x3 + -4x3 = 0 Combine like terms: -62x + -8x = -70x -107 + -70x + 2x2 + 20x2 + 4x3 + -4x3 = 0 Combine like terms: 2x2 + 20x2 = 22x2 -107 + -70x + 22x2 + 4x3 + -4x3 = 0 Combine like terms: 4x3 + -4x3 = 0 -107 + -70x + 22x2 + 0 = 0 -107 + -70x + 22x2 = 0 Solving -107 + -70x + 22x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 22 the coefficient of the squared term: Divide each side by '22'. -4.863636364 + -3.181818182x + x2 = 0 Move the constant term to the right: Add '4.863636364' to each side of the equation. -4.863636364 + -3.181818182x + 4.863636364 + x2 = 0 + 4.863636364 Reorder the terms: -4.863636364 + 4.863636364 + -3.181818182x + x2 = 0 + 4.863636364 Combine like terms: -4.863636364 + 4.863636364 = 0.000000000 0.000000000 + -3.181818182x + x2 = 0 + 4.863636364 -3.181818182x + x2 = 0 + 4.863636364 Combine like terms: 0 + 4.863636364 = 4.863636364 -3.181818182x + x2 = 4.863636364 The x term is -3.181818182x. Take half its coefficient (-1.590909091). Square it (2.530991736) and add it to both sides. Add '2.530991736' to each side of the equation. -3.181818182x + 2.530991736 + x2 = 4.863636364 + 2.530991736 Reorder the terms: 2.530991736 + -3.181818182x + x2 = 4.863636364 + 2.530991736 Combine like terms: 4.863636364 + 2.530991736 = 7.3946281 2.530991736 + -3.181818182x + x2 = 7.3946281 Factor a perfect square on the left side: (x + -1.590909091)(x + -1.590909091) = 7.3946281 Calculate the square root of the right side: 2.719306548 Break this problem into two subproblems by setting (x + -1.590909091) equal to 2.719306548 and -2.719306548.Subproblem 1
x + -1.590909091 = 2.719306548 Simplifying x + -1.590909091 = 2.719306548 Reorder the terms: -1.590909091 + x = 2.719306548 Solving -1.590909091 + x = 2.719306548 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.590909091' to each side of the equation. -1.590909091 + 1.590909091 + x = 2.719306548 + 1.590909091 Combine like terms: -1.590909091 + 1.590909091 = 0.000000000 0.000000000 + x = 2.719306548 + 1.590909091 x = 2.719306548 + 1.590909091 Combine like terms: 2.719306548 + 1.590909091 = 4.310215639 x = 4.310215639 Simplifying x = 4.310215639Subproblem 2
x + -1.590909091 = -2.719306548 Simplifying x + -1.590909091 = -2.719306548 Reorder the terms: -1.590909091 + x = -2.719306548 Solving -1.590909091 + x = -2.719306548 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.590909091' to each side of the equation. -1.590909091 + 1.590909091 + x = -2.719306548 + 1.590909091 Combine like terms: -1.590909091 + 1.590909091 = 0.000000000 0.000000000 + x = -2.719306548 + 1.590909091 x = -2.719306548 + 1.590909091 Combine like terms: -2.719306548 + 1.590909091 = -1.128397457 x = -1.128397457 Simplifying x = -1.128397457Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.310215639, -1.128397457}
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